Question: Solve for $z$. Reduce any fractions to lowest terms. Don't round your answer, and don't use mixed fractions. $-4z+31 \geq 17z + 23$
Explanation: $\begin{aligned}-4z+31 & \geq 17z + 23 \\\\ -4z&\geq 17z-8 &(\text{Subtract } 31 \text{ from both sides}) \\\\ -21z &\geq -8 &(\text{Subtract } 17z \text{ from both sides})\\\\ 21z&\leq 8&(\text{Multiply both sides by }-1)\\\\ z&\leq\dfrac{8}{21}&(\text{Divide both sides by }21) \end{aligned}$ [Why did the inequality sign flip when we multiplied by -1?] In conclusion, the answer is $z \leq \dfrac{8}{21}$.